In this article, we consider the asymptotic stability of the two-dimensional Boussinesq equations with partial dissipation near a combination of Couette flow and temperature profiles T(y). As a first main result, we show that if \(T'\) is of size at most \(\nu ^{1/3}\) in a suitable norm, then the linearized Boussinesq equations with only vertical dissipation of the velocity but not of the temperature are stable. Thus, mixing enhanced dissipation can suppress Rayleigh–Bénard instability in this linearized case. We further show that these results extend to the (forced) nonlinear equations with vertical dissipation in both temperature and velocity.

在本文中，我们考虑了在Couette流量和温度曲线T（y）的组合附近具有部分耗散的二维Boussinesq方程的渐近稳定性。作为第一个主要结果，我们证明如果在适当的范数下\（T'\）的大小最大为\（\ nu ^ {1/3} \），则线性化的Boussinesq方程仅具有速度的垂直耗散但温度不稳定。因此，在这种线性化情况下，混合增强的耗散可以抑制Rayleigh-Bénard的不稳定性。我们进一步表明，这些结果扩展到在温度和速度上都具有垂直耗散的（强迫）非线性方程。